feat: fit Ebbinghaus and exponential forgetting curves to recall@T data (#19)

Adds fit_forgetting_curve() to evaluation/stats.py — fits both exponential
(log-linear OLS) and Ebbinghaus (grid-search over S) models to recall@T
time-series and reports half-life, stability constant, and R² per backend.

CLI: --fit-curves flag prints the analysis after any benchmark run, works
with both single-seed and multi-seed (uses mean recall).

Merge resolves conflict with PR #18 --scenario arg; both args preserved.

Co-authored-by: Priyanshu-byte-coder
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

Author: Neal006

evaluation/stats.py

@@ -1,12 +1,20 @@
 """
-evaluation/stats.py — Statistical helpers for multi-seed aggregation.
+evaluation/stats.py — Statistical helpers for multi-seed aggregation and
+forgetting-curve analysis.
 
-Provides mean ± std + 95% confidence intervals over multiple benchmark runs.
+Provides:
+  - mean ± std + 95% confidence intervals over multiple benchmark runs
+  - fit_forgetting_curve() — fits Ebbinghaus / exponential decay to recall@T
+    data and returns half-life, stability, and R² goodness-of-fit
+
+Closes #6.
 """
 
 import math
-from typing import List, Optional, Tuple
+from typing import Dict, List, Optional, Tuple
+
 
+# ── Basic statistics ──────────────────────────────────────────────────────────
 
 def _mean(values: List[float]) -> float:
     return sum(values) / len(values) if values else 0.0
@@ -72,3 +80,172 @@ def aggregate_checkpoint_series(
         aggregate_metric([run[i] for run in series])
         for i in range(n_checkpoints)
     ]
+
+
+# ── Forgetting-curve fitting ──────────────────────────────────────────────────
+
+def _r_squared(observed: List[float], predicted: List[float]) -> float:
+    """Coefficient of determination R² ∈ (-∞, 1]; 1 = perfect fit."""
+    if len(observed) < 2:
+        return float("nan")
+    mean_obs = _mean(observed)
+    ss_tot = sum((y - mean_obs) ** 2 for y in observed)
+    ss_res = sum((y - y_hat) ** 2 for y, y_hat in zip(observed, predicted))
+    if ss_tot == 0:
+        return 1.0 if ss_res == 0 else float("-inf")
+    return 1.0 - ss_res / ss_tot
+
+
+def _fit_exponential(
+    turns: List[float],
+    recalls: List[float],
+) -> Tuple[float, float, float]:
+    """
+    Fit R(t) = a · exp(−k · t) via log-linear least squares.
+
+    Returns (a, k, r_squared).
+    a  — intercept (recall at t=0, ideally ≈ 1.0)
+    k  — decay rate (higher = faster forgetting)
+    """
+    # Filter out zero/negative recalls to avoid log(0)
+    valid = [(t, r) for t, r in zip(turns, recalls) if r > 0]
+    if len(valid) < 2:
+        return (float("nan"), float("nan"), float("nan"))
+
+    xs = [t for t, _ in valid]
+    ys = [math.log(r) for _, r in valid]
+
+    # Linear regression on log(R) = log(a) - k*t
+    n    = len(xs)
+    sx   = sum(xs)
+    sy   = sum(ys)
+    sxx  = sum(x * x for x in xs)
+    sxy  = sum(x * y for x, y in zip(xs, ys))
+    denom = n * sxx - sx * sx
+    if denom == 0:
+        return (float("nan"), float("nan"), float("nan"))
+
+    k       = -(n * sxy - sx * sy) / denom
+    log_a   = (sy - (-k) * sx) / n  # using -k because slope is -k
+    a       = math.exp(log_a)
+
+    predicted = [a * math.exp(-k * t) for t in turns]
+    r2 = _r_squared(recalls, predicted)
+    return (a, k, r2)
+
+
+def _fit_ebbinghaus(
+    turns: List[float],
+    recalls: List[float],
+    t_max:   float,
+) -> Tuple[float, float]:
+    """
+    Fit R(t) = exp(−t_norm / (S · sqrt(1 + t_norm))) by grid-searching over S.
+    t_norm = t / t_max so that t ∈ [0, 1].
+
+    Returns (S, r_squared).
+    S — stability constant (higher = slower forgetting)
+    """
+    if len(turns) < 2:
+        return (float("nan"), float("nan"))
+
+    t_norm_list = [t / max(t_max, 1) for t in turns]
+
+    def _predict(s: float) -> List[float]:
+        result = []
+        for tn in t_norm_list:
+            if tn <= 0:
+                result.append(1.0)
+            else:
+                denom = s * math.sqrt(1.0 + tn)
+                result.append(math.exp(-tn / denom))
+        return result
+
+    best_s  = 1.0
+    best_r2 = float("-inf")
+
+    # Coarse + fine grid search over S ∈ [0.01, 20]
+    for s in [i * 0.1 for i in range(1, 201)]:
+        predicted = _predict(s)
+        r2 = _r_squared(recalls, predicted)
+        if not math.isnan(r2) and r2 > best_r2:
+            best_r2 = r2
+            best_s  = s
+
+    return (best_s, best_r2)
+
+
+def fit_forgetting_curve(
+    checkpoints: List[int],
+    recalls:     List[float],
+) -> Dict:
+    """
+    Fit forgetting-curve models to a backend's recall@T time-series and return
+    interpretable memory-stability statistics.
+
+    Models fitted
+    -------------
+    exponential : R(t) = a · exp(−k · t)
+                  Classic single-parameter decay (Jost 1897).
+    ebbinghaus  : R(t) = exp(−t_norm / (S · √(1 + t_norm)))
+                  Two-parameter Ebbinghaus (1885) forgetting curve.
+
+    Parameters
+    ----------
+    checkpoints : list of turn numbers at which recall was measured
+    recalls     : list of recall values ∈ [0, 1] corresponding to each checkpoint
+
+    Returns
+    -------
+    dict with keys:
+      exponential:
+        a         — initial recall estimate at t=0
+        k         — decay rate (nats per turn)
+        half_life — turns until recall halves (ln(2)/k)
+        r2        — R² goodness-of-fit
+      ebbinghaus:
+        stability — S parameter (higher = more stable memory)
+        half_life — turns until recall drops to 0.5
+        r2        — R² goodness-of-fit
+      checkpoints : input turns (echoed for convenience)
+      recalls     : input recalls (echoed for convenience)
+    """
+    if len(checkpoints) != len(recalls) or len(checkpoints) < 2:
+        return {"error": "Need at least 2 (checkpoint, recall) pairs."}
+
+    turns   = [float(t) for t in checkpoints]
+    t_max   = max(turns)
+
+    # ── Exponential fit ──────────────────────────────────────────────────────
+    a, k, r2_exp = _fit_exponential(turns, recalls)
+    half_life_exp = math.log(2) / k if (not math.isnan(k) and k > 0) else float("inf")
+
+    # ── Ebbinghaus fit ───────────────────────────────────────────────────────
+    S, r2_ebb = _fit_ebbinghaus(turns, recalls, t_max)
+
+    # Half-life for Ebbinghaus: solve exp(-tn / (S * sqrt(1+tn))) = 0.5
+    # Numerically: scan t_norm values
+    half_life_ebb = float("inf")
+    if not math.isnan(S):
+        for step in range(1, 10001):
+            tn = step / 100.0
+            val = math.exp(-tn / (S * math.sqrt(1.0 + tn)))
+            if val <= 0.5:
+                half_life_ebb = round(tn * t_max, 2)
+                break
+
+    return {
+        "exponential": {
+            "a":         round(a,             4) if not math.isnan(a) else None,
+            "k":         round(k,             6) if not math.isnan(k) else None,
+            "half_life": round(half_life_exp, 2) if not math.isinf(half_life_exp) else None,
+            "r2":        round(r2_exp,        4) if not math.isnan(r2_exp) else None,
+        },
+        "ebbinghaus": {
+            "stability": round(S,             4) if not math.isnan(S) else None,
+            "half_life": half_life_ebb if not math.isinf(half_life_ebb) else None,
+            "r2":        round(r2_ebb,        4) if not math.isnan(r2_ebb) else None,
+        },
+        "checkpoints": checkpoints,
+        "recalls":     [round(r, 4) for r in recalls],
+    }

main.py

@@ -27,6 +27,10 @@
 Realistic chunked RAG backend:
     python main.py --backends naive rag_chunked cascading
 
+Forgetting-curve analysis (fit Ebbinghaus + exponential to recall@T data):
+    python main.py --fit-curves
+    python main.py --seeds 5 --fit-curves
+
 Other options:
     python main.py --turns 50 --backends naive rag --log
     python main.py --list-providers
@@ -66,6 +70,9 @@ def main() -> None:
     parser.add_argument("--decay",       type=str,  default="ebbinghaus",
                         choices=["ebbinghaus", "exponential", "linear", "default"],
                         help="Temporal decay function for CascadingMemory warm tier")
+    parser.add_argument("--fit-curves",  action="store_true",
+                        help="After benchmarking, fit Ebbinghaus + exponential decay curves "
+                             "to recall@T data and report half-life / stability / R²")
     parser.add_argument("--scenario",    type=str,  default="default",
                         choices=["default", "edtech"],
                         help="Conversation scenario: default (tech Q&A) | edtech (student-tutor)")
@@ -182,11 +189,51 @@ def main() -> None:
             })
             print(f"Experiment logged -> {path}")
 
+    # ── Forgetting-curve analysis ─────────────────────────────────────────────
+    if args.fit_curves:
+        from evaluation.stats import fit_forgetting_curve
+        checkpoints = sorted(args.checkpoints)
+        print("\nFORGETTING CURVE FIT  (Ebbinghaus + Exponential)")
+        print("-" * 65)
+        if multi_seed:
+            for name in args.backends:
+                if name not in aggregated:
+                    continue
+                mean_recalls = [stat["mean"] for stat in aggregated[name]["recall"]]
+                fit = fit_forgetting_curve(checkpoints, mean_recalls)
+                _print_curve_fit(name, fit)
+        else:
+            for name in args.backends:
+                if name not in display:
+                    continue
+                fit = fit_forgetting_curve(checkpoints, display[name]["recall"])
+                _print_curve_fit(name, fit)
+
     print("Visualise: streamlit run dashboard.py")
 
 
 # ── Output helpers ────────────────────────────────────────────────────────────
 
+
+def _print_curve_fit(backend: str, fit: dict) -> None:
+    if "error" in fit:
+        print(f"  {backend:<14}  {fit['error']}")
+        return
+    exp = fit["exponential"]
+    ebb = fit["ebbinghaus"]
+    hl_exp = f"{exp['half_life']:.1f} turns" if exp["half_life"] is not None else "N/A"
+    hl_ebb = f"{ebb['half_life']:.1f} turns" if ebb["half_life"] is not None else "N/A"
+    r2_exp = f"{exp['r2']:.3f}"              if exp["r2"]        is not None else "N/A"
+    r2_ebb = f"{ebb['r2']:.3f}"             if ebb["r2"]        is not None else "N/A"
+    stab   = f"{ebb['stability']:.4f}"       if ebb["stability"] is not None else "N/A"
+    k_val  = f"{exp['k']:.6f}"              if exp["k"]         is not None else "N/A"
+    print(f"  {backend}")
+    print(f"    Exponential  k={k_val}  half-life={hl_exp}  R²={r2_exp}")
+    print(f"    Ebbinghaus   S={stab}   half-life={hl_ebb}  R²={r2_ebb}")
+
+
+
+
 def _print_single_seed_results(display: dict, backends: list) -> None:
     checkpoints = display["checkpoints"]
     col = "  ".join(f"T={c:3d}" for c in checkpoints)